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Algebraic transformation of differential characteristic decompositions from one ranking to another. (English) Zbl 1174.13036

The authors propose an algorithm to transform a characteristic decomposition of a radical differential ideal from one ranking into another. In section 6 it is proved than the order of the elements of the canonical characteristic set of a characterizable differential ideal \(I\) is bounded by the order of \(I\). The bound is first proved for prime ideals and extended via characterizable ideals. This bound is used to compute a new bound for the order of an ordinary prime differential ideal in terms of a characteristic set w.r.t. a ranking \(\leq\), and this bound also works for the order of any derivative ocurring in the canonical characteristic set of the ideal w.r.t another ranking \(\leq'\).
This new bound does not depend of the ranking, and it is used to compute a canonical characteristic set \({\mathcal B}\) of a prime differential ideal w.r.t. the target ranking \(\leq'\) from a characteristic set \({\mathcal C}\) w.r.t. the input ranking \(\leq\). \({\mathcal B}\) is obtained as the weak d-triangular subset of \({\mathcal D}\) of least rank, where \({\mathcal D}\) is the canonical algebraic characteristic set of an algebraic ideal containing \({\mathcal C}\).
Previous idea is developed for differential ideals introducing differential prolongations. For a given differential ideal a differential prolongation is computed and a bi-characteristic decomposition \({\mathfrak D}\) of this differential prolongation is provided w.r.t. the input \(\leq\) and the target \(\leq'\) rankings. Then the characteristic decomposition of the original ideal is obtained as the minimal d-triangular subset \({\mathfrak C}\) of \({\mathfrak D}\).
Finally, since the proposed definition of canonical characteristic set does not agree with the original one by Boulier and Lemaire, the authors prove that their definition is correct. In the last section there is an algorithm to compute the canonical characteristic set of a characterizable radical differential ideal given by a finite set of differential polynomials.

MSC:

13N99 Differential algebra
13P99 Computational aspects and applications of commutative rings

Software:

RegularChains; Maple
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Full Text: DOI arXiv

References:

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